Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $k \neq 0$. $a = \dfrac{7k(4k - 1)}{3} \div \dfrac{10(4k - 1)}{10} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{7k(4k - 1)}{3} \times \dfrac{10}{10(4k - 1)} $ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ 7k(4k - 1) \times 10 } { 3 \times 10(4k - 1) } $ $ a = \dfrac{70k(4k - 1)}{30(4k - 1)} $ We can cancel the $4k - 1$ so long as $4k - 1 \neq 0$ Therefore $k \neq \dfrac{1}{4}$ $a = \dfrac{70k \cancel{(4k - 1})}{30 \cancel{(4k - 1)}} = \dfrac{70k}{30} = \dfrac{7k}{3} $